The top-four sides took turns dominating Thursday night's clash in Darwin, the Hawks leading in the final quarter after going behind by 32 points in the second term.
But the Suns - chasing a maiden finals appearance - had the answers in a 16.8 (104) to 15.6 (96) win that jumped them ahead of the Hawks into second on the live ladder with a 7-2 record.
Miller (22 disposals, eight clearances, two goals) popped up with a major that pushed the lead to eight points late in the final quarter.
Northern Territory product Ben Long (four goals), halfback John Noble (36 disposals) and captain Noah Anderson (35, 12 clearances, one goal) starred as they made it 8-0 at the venue.
Anderson led the Suns' first-half assault out of the middle, winning nine clearances to often allow his side direct entry to the forward line.
Noble had 22 touches in the first half, his influence from the backline enough for Hawks coach Sam Mitchell to dedicate a man to him.
Their impact meant the Suns piled on quick goals, Miller kicking their first, Long bagging a couple and six others getting on the score sheet.
Sam Flanders and Anderson himself both kicked goals to establish 32-point margins in the second term.
But pocket rocket Nick Watson almost single-handedly kept the Hawks in it.
He threaded the needle from the pocket, outmarked the taller Wil Powell to kick another and then won a turnover and banged a third home from long-range.
Jack Ginnivan chimed in and the lead was back to 14 at halftime.
Their midfield lifted and it was the Hawks who controlled things in the third term as tempers flared and the pressure lifted.
Hawthorn kicked eight of 10 goals either side of halftime to slowly peg back the margin and go ahead seconds before the final break when Blake Hardwick kicked his third.
Matt Rowell (29 touches, nine clearances) lifted in the final quarter, his effort at the contest leading to Malcolm Rosas Jnr's runaway goal that put the Suns 15 points clear with 10 minutes to play.
The Hawks pressed again Ginnivan making no mistake after being taken high by Nick Holman and kicking truly to make it a two-point game.
Miller popped up again though for a crucial second goal before Rowell and Anderson combined to assist Long's fourth.
Chol was brilliant with a one-handed pick-up and cool finish for his third goal that made it a eight-point game with three minutes to play.
But, like they did five nights ago against the Western Bulldogs, the Suns showed poise and scramble to hang on.
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